Acoustic modes of a rigid cylindrical cavity (${\omega }^2=c_0^2[(k_{m,n}/a{)}^2+(\ell \pi /h{)}^2]$)

For a rigid cylinder of radius $a$ and height $h$, the velocity potential obeys the wave equation with Neumann boundary conditions. Separation in cylindrical coordinates gives radial factors $J_m(k_{m,n}r/a)$, angular sine and cosine factors of integer order $m$, and axial factors $\cos (\ell \pi z/h)$. The nonzero radial numbers satisfy $J_m^{\prime }(k_{m,n})=0$, yielding the displayed frequencies.

There is also a constant radial mode, with $m=0$ and radial number zero. A formal zero of $J_m^{\prime }$ at the origin for higher $m$ must not be counted by substituting $k=0$ into $J_m(kr)$, which would give the zero function. Solving the zero-radial-eigenvalue equation directly leaves only the constant mode under the side-wall condition. The fully constant potential has zero velocity and is a gauge mode; constant radial modes with nonzero axial index are physical longitudinal oscillations.